UniformManifoldGeometry.project
===============================

.. py:method:: t3toolbox.uniform_manifold.UniformManifoldGeometry.project(v)

   .. code-block:: python

      def project(
              self,
              v:  UT3Tangent,
      ) -> UT3Tangent:  # gauged tangent at v's frame (a DIFFERENT vector)


   The gauge projection ``Pi``: raw cotangent -> Riemannian gradient (orthogonal gauge).

   Orthogonally projects ``v``'s variations onto the gauged tangent space (conditions (48)-(49),
   Appendix A.3). Represents a DIFFERENT tangent vector than ``v`` -- the map that turns a bare adjoint
   ``J^T r`` into the Riemannian gradient. For the vector-preserving fix see :py:meth:`project_oblique`.
   **Safe mode** requires ``v``'s frame orthogonal (raises otherwise; skipped under ``safety.unsafe()`` /
   a jax trace).